Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol24.Issue1.Jan2000:
Journal of Economic Dynamics & Control
24 (2000) 39}62
Indeterminate growth paths and stability
Thomas Russell!,*, Aleksandar Zecevic"
! Department of Economics, Santa Clara University, Santa Clara, CA 95053, USA
" Department of Electrical Engineering, Santa Clara University, Santa Clara, CA 95053, USA
Received 6 January 1997; accepted 10 August 1998
Abstract
Indeterminacy in an economic growth model arises whenever the stable manifold has
dimension greater than the number of predetermined initial conditions. The stability
(indeterminacy) of transition paths in the Benhabib and Farmer (1996) model of growth is
investigated, using both the Lyapunov method and numerical simulation techniques. The
sensitivity of transient dynamics is analyzed with respect to the choice of parameter
values and with respect to the choice of initial conditions. The likelihood that the
business cycle is a pure &sunspot' phenomenon is investigated. ( 2000 Elsevier Science
B.V. All rights reserved.
JEL classixcation: E00; E3; O40
Keywords: Indeterminate growth; Lyapunov stability; Sunspot equilibrium
1. Introduction
Why does one economy grow faster than another? This remains one of the key
questions of economic science, in large part because of the hope that an
understanding of the causes of growth will point to policies which will enable
countries to achieve faster growth and therefore higher standards of living.
It seems very natural to begin the search for an explanation of why growth
rates di!er by looking for di!erences across countries in those fundamental
economic attributes which might be expected to contribute to higher growth.
For example, countries with higher growth rates may save more. Or, such
* Corresponding author. E-mail: [email protected]
0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 8 ) 0 0 0 6 0 - 8
40
T. Russell, A. Zecevic / Journal of Economic Dynamics & Control 24 (2000) 39}62
countries may have access to superior technology. Or perhaps citizens of higher
growth countries simply work harder.
This list of fundamental economic attributes could be expanded, but recently
a number of studies have called this &fundamentalist' approach into question. In
a variety of models of growth, there are empirically reasonable parameter values
at which economic growth is indeterminate. That is to say, for given preferences,
given technology, and a given initial capital stock, the future equilibrium growth
path of the economy is not unique, indeed is not even locally unique. This means
that two &fundamentally' identical economies could evolve along quite di!erent
growth paths.
The possibility of indeterminacy arises in a variety of economic growth
models. For example, the Lucas (1988) model of economic growth with human
capital externalities is not uniquely determined by specifying initial physical and
human capital, as shown in Xie (1994). Models with physical capital and
externalities have been investigated by, among others, Matsuyama (1991),
Boldrin (1992), Boldrin and Rustichini (1994), Benhabib and Farmer (1994),
Benhabib and Perli (1994), Gali (1994) and Benhabib and Farmer (1996).
Indeterminacy in one sector models has also been considered by a number of
authors, including Kehoe et al. (1991), Kehoe (1991) and Spear (1991).
In addition to its implications for growth theory, indeterminacy also has
implications for the study of the business cycle. When the equilibrium is
indeterminate, &animal spirits' can generate belief-driven cycles in output even
when the economy has no underlying fundamental (e.g. technological) uncertainty (see Farmer and Guo, 1994). A clear statement of these issues may be
found in the symposium introduction by Benhabib and Rustichini (1994).
Many of the models of indeterminacy discussed in this symposium and
elsewhere have a common mathematical structure. An economy's equilibrium
growth path is represented as a dynamic system, typically a second-order
nonlinear di!erential (or di!erence) equation. Next, the steady state (or steady
growth path) of this dynamic system is investigated. When the steady state of
such a system is locally stable, (i.e. when the eigenvalues of the Jacobian of the
system at the steady state lie in the left half plane) indeterminacy follows. To see
this, suppose we specify one initial condition, say the initial capital stock, K .
0
Because of stability, there is now an open set of values of the second initial
condition, say initial consumption, such that any choice of initial consumption
in this set, taken in combination with K , converges to the steady state. Because
0
the model contains no explanation for which level of initial consumption will be
chosen, growth is said to be &indeterminate'. Furthermore, when such a system is
subject to an expectations shock, it produces a pure &animal spirits' cycle.
The fact that an equilibrium system is indeterminate is clearly of great
theoretical interest. What is not so clear is how much light this result sheds on
the diversity of experience of actual economies. Negative eigenvalues of the
Jacobian at the steady state give information about the behavior of the system
T. Russell, A. Zecevic / Journal of Economic Dynamics & Control 24 (2000) 39}62
41
only in a neighborhood of its steady state. This neighborhood must be chosen
small enough that linearization approximations hold, and so may be very small
indeed.
Ideally, we seek information on the behavior of indeterminate economies over
wide ranges of initial conditions and for long periods of time. Moreover, it
would be interesting to see how such economies behave when their dynamics is
not constrained to be linear. To obtain such information, di!erent mathematical
techniques must be used.
In this paper we analyze the indeterminacy issue using two approaches.
Firstly, we apply the classical stability analysis of Lyapunov to obtain an
analytical estimate of the set of stable (and therefore indeterminate) trajectories.
By exhibiting a Lyapunov function for an indeterminate growth model, we can
explicitly evaluate the transitional dynamics for two economies with the same
initial conditions. This allows us to examine whether or not stable models
produce dynamic paths consistent with the stylized facts of comparative growth
experience as set out in, say, Barro and Sala-i-Martin (1995). It also allows us to
bound the range of behavior attributable to &animal spirits'. Paths produced by
stochastic shocks in sunspot equilibrium models must converge to the models'
steady state, so by describing a Lyapunov region we shed some light on the
behavior of such paths.1
Even before we begin this task, it is clear that there is some reason to doubt
the empirical importance of Lyapunov stable indeterminate systems. Suppose
we have a two-dimensional nonlinear dynamic system, in which K(t) represents
the capital stock in the economy at time t and C(t) is the consumption level in
the economy at time t. In addition, let K* and C* be the common steady state
values of capital and consumption in the economy, and let x (t),log (K(t)/K*)
1
and x (t),log (C(t)/C*), respectively. Then, for Lyapunov stable indeterminate
2
systems, the following properties hold.
1. Given x(t),(x (t)x (t))T, Lyapunov stability (by de"nition) ensures that for
1 2
any number e'0 there exists a small enough d'0 (generally depending on
e) such that Ex(t)E(e is guaranteed by Ex(0)E(d(e). In other words, economies which start o! close together remain close together at all times.
2. There exists a set C such that x(t)3C for all t50 and for any trajectory
originating in this set.
3. Two economies whose initial conditions belong to the set C will converge
uniformly to their common steady state.
4. Although two economies which are fundamentally equivalent may exhibit
di!erent growth behavior, they spend an in"nite amount of time on equilibrium growth paths which di!er from each other by less than g, where g is any
positive constant, no matter how small.
1 This interesting interpretation of our results was kindly provided by a referee.
42
T. Russell, A. Zecevic / Journal of Economic Dynamics & Control 24 (2000) 39}62
Since economic measurement is always imprecise, it may even be the case that
the indeterminacy predicted by Lyapunov's stability theory cannot be detected
empirically. Indeed, we will certainly be unable to detect indeterminacy for
in"nite amounts of time if we use measuring instruments which are calibrated in
"nite jumps.
This is not to say that indeterminacy in general is an empirically uninteresting
phenomenon. It is simply to point out that using either local stability theory
around the steady state or Lyapunov stability theorems as a way of detecting
indeterminacy only allows us to predict a form of indeterminacy that is so local
and so transient that we may never be able to observe it. Moreover, although the
"nding of local Lyapunov stability is su$cient to guarantee indeterminacy, it is
certainly not necessary. Indeterminacy of the equilibrium growth path of an
economic system will also follow if the steady state is an attractor (i.e. if a set of
paths tends to the common steady state). The set of initial conditions for which
the trajectories converge to the steady state will be referred to as the region of
attraction, and it is important to point out that this region need not coincide with
the set C.
This other form of indeterminacy (i.e. that associated with the region of
attraction) can yield much more varied growth experience for two fundamentally identical economies than that generated by Lyapunov stability analysis. For example, two &attractive' economies with the same initial conditions
could move along sharply divergent paths before coming together, behavior
which is not possible if the economies originate in the set C.
For that reason, the second purpose of this paper is to revisit the issue of
indeterminacy using numerical techniques. The only paper we are aware of
which does this in the indeterminate case is the paper by Xie (1994). However,
Xie was only able to solve the dynamic model explicitly by imposing restrictions
on the parameters which simplify the underlying di!erential equations. These
restrictions have no obvious empirical basis, so Xie could only speculate on the
nature of the dynamics with economically reasonable parameter values. In this
paper we will examine the explicit dynamics of equilibrium growth paths using
empirically reasonable parameter values within the context of one model that is
known to generate indeterminacy, the interesting externality model of Benhabib
and Farmer (1996) (which is an extension of the same authors' model from
1994).2
Our two methods of analysis are complementary. In the "rst place we will
exhibit a Lyapunov function for this model. For a given capital stock, this
function allows us to estimate a region of initial consumption levels for which
the model is stable in the sense of Lyapunov. This, in turn, allows us to explicitly
2 It is possible that the numerical techniques used by Mulligan and Sala-i-Martin (1993) could also
be adapted to deal with this model, but we have not pursued this task.
T. Russell, A. Zecevic / Journal of Economic Dynamics & Control 24 (2000) 39}62
43
bound both the transient equilibrium dynamics of economies which start in this
region and the set of sunspot equilibria. Since this is the range of uniform
convergence, indeterminacy in this region may have less empirical interest.
With that in mind, we will also use numerical techniques to extend the
Lyapunov estimates of indeterminacy to the entire region of attraction. Using
these techniques we try to answer a number of questions related to the empirical
importance of the indeterminacy issue. In particular, we are interested in
examining the size of the region of attraction, the behavior of paths originating
in this region, and the sensitivity of these paths to the choice of initial conditions
and parameters. These questions will be answered within the context of a wellknown &indeterminate' model which we now describe.
2. The Benhabib}Farmer model
The following model is due to Benhabib and Farmer (1996). It is a standard
Ramsay model of growth to which aggregate and sector speci"c externalities
have been added. The model consists of two sectors } a consumption sector
C and an investment sector I, with sector output produced by the respective
private technologies
C"A(k K)a(k ¸)b,
k
L
I"B[(1!k )K]a[(1!k )¸]b.
k
L
(1)
In Eq. (1) K represents the economy wide stock of capital, ¸ is the economy
wide stock of labor, and k and k are the respective fractions of K and ¸ used in
K
L
the consumption sector. Individual "rms take A and B to be constant, and
constant returns to scale hold at this level, implying a#b"1.
Externalities are introduced by assuming that
A"(kN KM )ah(kN ¸M )bhKM ap¸M bc,
B"[(1!kN )KM ]ah[(1!kN )¸M ]bhKM ap¸M bc, (2)
k
L
K
L
where a bar over a variable represents average economy wide use, and averages
are taken as given by the individual "rm. The parameter h is a measure of sector
speci"c externalities, while p and c are measures of aggregate capital and labor
external e!ects, respectively. Given these parameters, it is convenient to de"ne
three additional quantities: l,(1#h), a,a(1#h#p) and b,(1#h#c).
The consumer's preferences are given by a time separable utility functional
whose instantaneous value is given by
¸(t)1`s
,
;(t)"log C(t)!
1#s
(3)
where C(t) is consumption, ¸(t) is the labor supply and s represents the labor
elasticity parameter. The consumer's optimization problem is then to maximize
44
T. Russell, A. Zecevic / Journal of Economic Dynamics & Control 24 (2000) 39}62
the integral
P
=
;(t)e~ot dt
(4)
KQ (t)"I(t)!dK(t)
(5)
J"
0
subject to
and K(0)" K , where I(t) denotes investment goods and d is the depreciation
0
rate of capital.
The necessary conditions for optimizing integral can be formulated in terms
of its "rst variation. Namely, in order for pair MK(t), C(t)N to maximize it is
necessary that the "rst variation equals zero along this trajectory; it is also
required that the trajectory satis"es the transversality condition
lim e~otK(t)K(t)"0,
t?=
(6)
where K(t) is the standard co-state variable associated with the Hamiltonian
formulation of the optimization problem.
In analyzing the "rst variation of Eq. (4), it is convenient to introduce a new
variable S de"ned as
Ka@l¸b@l
S,
.
C1@l
(7)
The quantity S takes values between one and in"nity, and can be interpreted
as the inverse of the factor share going to the consumption sector. Based on
de"nition Eq. (7), we obtain relationships
bS"¸1`s,
1
C(S!1)l~1" ,
K
I"C(S!1)l
(8)
and the solution to the optimization problem reduces to a pair of di!erential
equations
KQ
S
"o#d!a
,
K
KK
KQ
S!1
"
!d.
K
KK
(9)
It is important to point out that variable S is in fact an implicit function of
K and K. Indeed, from Eqs. (7) and (8) one directly obtains
(S!1)1~lSl~(b@(1`s))"b(b@(1`s))KaK,
(10)
T. Russell, A. Zecevic / Journal of Economic Dynamics & Control 24 (2000) 39}62
45
which implies that the only independent variables in Eq. (9) are K and K. It is
also easily established that system (9) has a unique steady state, which can be
computed in the following sequence:
o#d
S*"
o#d(1!a)
N
o#d
K*"
(S*!1)(1~l)@(a~1)S**(l~1)@(a~1)~b@((a~1)(1`s))+bb@((a~1)(1`s))
a
aS*
1
N K*"
N C*" (S*!1)1~l.
(o#d)K*
K*
(11)
For the purposes of stability analysis, it will be convenient to work with
logarithmic variables j,log K and k,log K, and use them to de"ne
x ,j!j* and x ,k!k*. This produces a new system of di!erential equa1
2
tions
xR "o#d!awSe~x1~x2,
1
(12)
xR "wSe~x1~x2#we~x1~x2!o,
2
in which w,e~kH~jH"(o#d(1!a))/a, and the steady state is at the origin. In
other words, the problem can be formulated as a nonlinear dynamic system in
the form
xR "f (x),
f (0)"0
(13)
which is suitable for a Lyapunov-type stability analysis. We should also mention
that variable S can be expressed in terms of x and x as
1
2
(14)
(S!1)pSq"Meax2`x1,
where p,1!l, q,l!b/(1#s) and M,bb@(1`s)eakH`jH.
Eq. (13) can be simpli"ed by computing the Jacobian A(x) and linearizing
around the steady state, which produces
xR "A(0)x
(15)
As is well known, the linear system (15) will be stable if and only if A(0) has
both eigenvalues in the left half of the complex plane. The necessary and
su$cient conditions for this can be stated explicitly in terms of the trace and
determinant of A(0)
RS
o#d(1!a)
(a!a)
(0,
Trace"o#
Rx
a
1
[o#d(1!a)]2
RS
Det"
(a!1)
'0,
Rx
a
1
(16)
46
T. Russell, A. Zecevic / Journal of Economic Dynamics & Control 24 (2000) 39}62
where RS/Rx is evaluated at the steady state using Eq. (14). Such a formulation
1
allows for an easy identi"cation of parameters for which indeterminacy can
occur. We should point out, however, that the stability of Eq. (15) only guarantees local stability of Eq. (13) around the origin. In other words, since Eq. (15) is
a local approximation, all that can be said based on the eigenvalues of the
Jacobian is that indeterminacy exists in a neighborhood of the origin.
3. The Lyapunov method
To obtain an explicit bound on initial conditions that lead to indeterminacy,
it is necessary to replace linearization with a more sophisticated approach. In
this section we will use Lyapunov's method to estimate a region of stability for
this model. The following well known result provides a basis for our analysis
(e.g. Rouche et al., 1977).
Theorem 1. Let x(t;x ) denote the solution of the nonlinear dynamic system (12)
0
that corresponds to initial condition x(0)"x , and let XLRn be a set contain0
ing the origin. Assume also that there exists a continuously di!erentiable
function < : XPR satisfying
24 (2000) 39}62
Indeterminate growth paths and stability
Thomas Russell!,*, Aleksandar Zecevic"
! Department of Economics, Santa Clara University, Santa Clara, CA 95053, USA
" Department of Electrical Engineering, Santa Clara University, Santa Clara, CA 95053, USA
Received 6 January 1997; accepted 10 August 1998
Abstract
Indeterminacy in an economic growth model arises whenever the stable manifold has
dimension greater than the number of predetermined initial conditions. The stability
(indeterminacy) of transition paths in the Benhabib and Farmer (1996) model of growth is
investigated, using both the Lyapunov method and numerical simulation techniques. The
sensitivity of transient dynamics is analyzed with respect to the choice of parameter
values and with respect to the choice of initial conditions. The likelihood that the
business cycle is a pure &sunspot' phenomenon is investigated. ( 2000 Elsevier Science
B.V. All rights reserved.
JEL classixcation: E00; E3; O40
Keywords: Indeterminate growth; Lyapunov stability; Sunspot equilibrium
1. Introduction
Why does one economy grow faster than another? This remains one of the key
questions of economic science, in large part because of the hope that an
understanding of the causes of growth will point to policies which will enable
countries to achieve faster growth and therefore higher standards of living.
It seems very natural to begin the search for an explanation of why growth
rates di!er by looking for di!erences across countries in those fundamental
economic attributes which might be expected to contribute to higher growth.
For example, countries with higher growth rates may save more. Or, such
* Corresponding author. E-mail: [email protected]
0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 8 ) 0 0 0 6 0 - 8
40
T. Russell, A. Zecevic / Journal of Economic Dynamics & Control 24 (2000) 39}62
countries may have access to superior technology. Or perhaps citizens of higher
growth countries simply work harder.
This list of fundamental economic attributes could be expanded, but recently
a number of studies have called this &fundamentalist' approach into question. In
a variety of models of growth, there are empirically reasonable parameter values
at which economic growth is indeterminate. That is to say, for given preferences,
given technology, and a given initial capital stock, the future equilibrium growth
path of the economy is not unique, indeed is not even locally unique. This means
that two &fundamentally' identical economies could evolve along quite di!erent
growth paths.
The possibility of indeterminacy arises in a variety of economic growth
models. For example, the Lucas (1988) model of economic growth with human
capital externalities is not uniquely determined by specifying initial physical and
human capital, as shown in Xie (1994). Models with physical capital and
externalities have been investigated by, among others, Matsuyama (1991),
Boldrin (1992), Boldrin and Rustichini (1994), Benhabib and Farmer (1994),
Benhabib and Perli (1994), Gali (1994) and Benhabib and Farmer (1996).
Indeterminacy in one sector models has also been considered by a number of
authors, including Kehoe et al. (1991), Kehoe (1991) and Spear (1991).
In addition to its implications for growth theory, indeterminacy also has
implications for the study of the business cycle. When the equilibrium is
indeterminate, &animal spirits' can generate belief-driven cycles in output even
when the economy has no underlying fundamental (e.g. technological) uncertainty (see Farmer and Guo, 1994). A clear statement of these issues may be
found in the symposium introduction by Benhabib and Rustichini (1994).
Many of the models of indeterminacy discussed in this symposium and
elsewhere have a common mathematical structure. An economy's equilibrium
growth path is represented as a dynamic system, typically a second-order
nonlinear di!erential (or di!erence) equation. Next, the steady state (or steady
growth path) of this dynamic system is investigated. When the steady state of
such a system is locally stable, (i.e. when the eigenvalues of the Jacobian of the
system at the steady state lie in the left half plane) indeterminacy follows. To see
this, suppose we specify one initial condition, say the initial capital stock, K .
0
Because of stability, there is now an open set of values of the second initial
condition, say initial consumption, such that any choice of initial consumption
in this set, taken in combination with K , converges to the steady state. Because
0
the model contains no explanation for which level of initial consumption will be
chosen, growth is said to be &indeterminate'. Furthermore, when such a system is
subject to an expectations shock, it produces a pure &animal spirits' cycle.
The fact that an equilibrium system is indeterminate is clearly of great
theoretical interest. What is not so clear is how much light this result sheds on
the diversity of experience of actual economies. Negative eigenvalues of the
Jacobian at the steady state give information about the behavior of the system
T. Russell, A. Zecevic / Journal of Economic Dynamics & Control 24 (2000) 39}62
41
only in a neighborhood of its steady state. This neighborhood must be chosen
small enough that linearization approximations hold, and so may be very small
indeed.
Ideally, we seek information on the behavior of indeterminate economies over
wide ranges of initial conditions and for long periods of time. Moreover, it
would be interesting to see how such economies behave when their dynamics is
not constrained to be linear. To obtain such information, di!erent mathematical
techniques must be used.
In this paper we analyze the indeterminacy issue using two approaches.
Firstly, we apply the classical stability analysis of Lyapunov to obtain an
analytical estimate of the set of stable (and therefore indeterminate) trajectories.
By exhibiting a Lyapunov function for an indeterminate growth model, we can
explicitly evaluate the transitional dynamics for two economies with the same
initial conditions. This allows us to examine whether or not stable models
produce dynamic paths consistent with the stylized facts of comparative growth
experience as set out in, say, Barro and Sala-i-Martin (1995). It also allows us to
bound the range of behavior attributable to &animal spirits'. Paths produced by
stochastic shocks in sunspot equilibrium models must converge to the models'
steady state, so by describing a Lyapunov region we shed some light on the
behavior of such paths.1
Even before we begin this task, it is clear that there is some reason to doubt
the empirical importance of Lyapunov stable indeterminate systems. Suppose
we have a two-dimensional nonlinear dynamic system, in which K(t) represents
the capital stock in the economy at time t and C(t) is the consumption level in
the economy at time t. In addition, let K* and C* be the common steady state
values of capital and consumption in the economy, and let x (t),log (K(t)/K*)
1
and x (t),log (C(t)/C*), respectively. Then, for Lyapunov stable indeterminate
2
systems, the following properties hold.
1. Given x(t),(x (t)x (t))T, Lyapunov stability (by de"nition) ensures that for
1 2
any number e'0 there exists a small enough d'0 (generally depending on
e) such that Ex(t)E(e is guaranteed by Ex(0)E(d(e). In other words, economies which start o! close together remain close together at all times.
2. There exists a set C such that x(t)3C for all t50 and for any trajectory
originating in this set.
3. Two economies whose initial conditions belong to the set C will converge
uniformly to their common steady state.
4. Although two economies which are fundamentally equivalent may exhibit
di!erent growth behavior, they spend an in"nite amount of time on equilibrium growth paths which di!er from each other by less than g, where g is any
positive constant, no matter how small.
1 This interesting interpretation of our results was kindly provided by a referee.
42
T. Russell, A. Zecevic / Journal of Economic Dynamics & Control 24 (2000) 39}62
Since economic measurement is always imprecise, it may even be the case that
the indeterminacy predicted by Lyapunov's stability theory cannot be detected
empirically. Indeed, we will certainly be unable to detect indeterminacy for
in"nite amounts of time if we use measuring instruments which are calibrated in
"nite jumps.
This is not to say that indeterminacy in general is an empirically uninteresting
phenomenon. It is simply to point out that using either local stability theory
around the steady state or Lyapunov stability theorems as a way of detecting
indeterminacy only allows us to predict a form of indeterminacy that is so local
and so transient that we may never be able to observe it. Moreover, although the
"nding of local Lyapunov stability is su$cient to guarantee indeterminacy, it is
certainly not necessary. Indeterminacy of the equilibrium growth path of an
economic system will also follow if the steady state is an attractor (i.e. if a set of
paths tends to the common steady state). The set of initial conditions for which
the trajectories converge to the steady state will be referred to as the region of
attraction, and it is important to point out that this region need not coincide with
the set C.
This other form of indeterminacy (i.e. that associated with the region of
attraction) can yield much more varied growth experience for two fundamentally identical economies than that generated by Lyapunov stability analysis. For example, two &attractive' economies with the same initial conditions
could move along sharply divergent paths before coming together, behavior
which is not possible if the economies originate in the set C.
For that reason, the second purpose of this paper is to revisit the issue of
indeterminacy using numerical techniques. The only paper we are aware of
which does this in the indeterminate case is the paper by Xie (1994). However,
Xie was only able to solve the dynamic model explicitly by imposing restrictions
on the parameters which simplify the underlying di!erential equations. These
restrictions have no obvious empirical basis, so Xie could only speculate on the
nature of the dynamics with economically reasonable parameter values. In this
paper we will examine the explicit dynamics of equilibrium growth paths using
empirically reasonable parameter values within the context of one model that is
known to generate indeterminacy, the interesting externality model of Benhabib
and Farmer (1996) (which is an extension of the same authors' model from
1994).2
Our two methods of analysis are complementary. In the "rst place we will
exhibit a Lyapunov function for this model. For a given capital stock, this
function allows us to estimate a region of initial consumption levels for which
the model is stable in the sense of Lyapunov. This, in turn, allows us to explicitly
2 It is possible that the numerical techniques used by Mulligan and Sala-i-Martin (1993) could also
be adapted to deal with this model, but we have not pursued this task.
T. Russell, A. Zecevic / Journal of Economic Dynamics & Control 24 (2000) 39}62
43
bound both the transient equilibrium dynamics of economies which start in this
region and the set of sunspot equilibria. Since this is the range of uniform
convergence, indeterminacy in this region may have less empirical interest.
With that in mind, we will also use numerical techniques to extend the
Lyapunov estimates of indeterminacy to the entire region of attraction. Using
these techniques we try to answer a number of questions related to the empirical
importance of the indeterminacy issue. In particular, we are interested in
examining the size of the region of attraction, the behavior of paths originating
in this region, and the sensitivity of these paths to the choice of initial conditions
and parameters. These questions will be answered within the context of a wellknown &indeterminate' model which we now describe.
2. The Benhabib}Farmer model
The following model is due to Benhabib and Farmer (1996). It is a standard
Ramsay model of growth to which aggregate and sector speci"c externalities
have been added. The model consists of two sectors } a consumption sector
C and an investment sector I, with sector output produced by the respective
private technologies
C"A(k K)a(k ¸)b,
k
L
I"B[(1!k )K]a[(1!k )¸]b.
k
L
(1)
In Eq. (1) K represents the economy wide stock of capital, ¸ is the economy
wide stock of labor, and k and k are the respective fractions of K and ¸ used in
K
L
the consumption sector. Individual "rms take A and B to be constant, and
constant returns to scale hold at this level, implying a#b"1.
Externalities are introduced by assuming that
A"(kN KM )ah(kN ¸M )bhKM ap¸M bc,
B"[(1!kN )KM ]ah[(1!kN )¸M ]bhKM ap¸M bc, (2)
k
L
K
L
where a bar over a variable represents average economy wide use, and averages
are taken as given by the individual "rm. The parameter h is a measure of sector
speci"c externalities, while p and c are measures of aggregate capital and labor
external e!ects, respectively. Given these parameters, it is convenient to de"ne
three additional quantities: l,(1#h), a,a(1#h#p) and b,(1#h#c).
The consumer's preferences are given by a time separable utility functional
whose instantaneous value is given by
¸(t)1`s
,
;(t)"log C(t)!
1#s
(3)
where C(t) is consumption, ¸(t) is the labor supply and s represents the labor
elasticity parameter. The consumer's optimization problem is then to maximize
44
T. Russell, A. Zecevic / Journal of Economic Dynamics & Control 24 (2000) 39}62
the integral
P
=
;(t)e~ot dt
(4)
KQ (t)"I(t)!dK(t)
(5)
J"
0
subject to
and K(0)" K , where I(t) denotes investment goods and d is the depreciation
0
rate of capital.
The necessary conditions for optimizing integral can be formulated in terms
of its "rst variation. Namely, in order for pair MK(t), C(t)N to maximize it is
necessary that the "rst variation equals zero along this trajectory; it is also
required that the trajectory satis"es the transversality condition
lim e~otK(t)K(t)"0,
t?=
(6)
where K(t) is the standard co-state variable associated with the Hamiltonian
formulation of the optimization problem.
In analyzing the "rst variation of Eq. (4), it is convenient to introduce a new
variable S de"ned as
Ka@l¸b@l
S,
.
C1@l
(7)
The quantity S takes values between one and in"nity, and can be interpreted
as the inverse of the factor share going to the consumption sector. Based on
de"nition Eq. (7), we obtain relationships
bS"¸1`s,
1
C(S!1)l~1" ,
K
I"C(S!1)l
(8)
and the solution to the optimization problem reduces to a pair of di!erential
equations
KQ
S
"o#d!a
,
K
KK
KQ
S!1
"
!d.
K
KK
(9)
It is important to point out that variable S is in fact an implicit function of
K and K. Indeed, from Eqs. (7) and (8) one directly obtains
(S!1)1~lSl~(b@(1`s))"b(b@(1`s))KaK,
(10)
T. Russell, A. Zecevic / Journal of Economic Dynamics & Control 24 (2000) 39}62
45
which implies that the only independent variables in Eq. (9) are K and K. It is
also easily established that system (9) has a unique steady state, which can be
computed in the following sequence:
o#d
S*"
o#d(1!a)
N
o#d
K*"
(S*!1)(1~l)@(a~1)S**(l~1)@(a~1)~b@((a~1)(1`s))+bb@((a~1)(1`s))
a
aS*
1
N K*"
N C*" (S*!1)1~l.
(o#d)K*
K*
(11)
For the purposes of stability analysis, it will be convenient to work with
logarithmic variables j,log K and k,log K, and use them to de"ne
x ,j!j* and x ,k!k*. This produces a new system of di!erential equa1
2
tions
xR "o#d!awSe~x1~x2,
1
(12)
xR "wSe~x1~x2#we~x1~x2!o,
2
in which w,e~kH~jH"(o#d(1!a))/a, and the steady state is at the origin. In
other words, the problem can be formulated as a nonlinear dynamic system in
the form
xR "f (x),
f (0)"0
(13)
which is suitable for a Lyapunov-type stability analysis. We should also mention
that variable S can be expressed in terms of x and x as
1
2
(14)
(S!1)pSq"Meax2`x1,
where p,1!l, q,l!b/(1#s) and M,bb@(1`s)eakH`jH.
Eq. (13) can be simpli"ed by computing the Jacobian A(x) and linearizing
around the steady state, which produces
xR "A(0)x
(15)
As is well known, the linear system (15) will be stable if and only if A(0) has
both eigenvalues in the left half of the complex plane. The necessary and
su$cient conditions for this can be stated explicitly in terms of the trace and
determinant of A(0)
RS
o#d(1!a)
(a!a)
(0,
Trace"o#
Rx
a
1
[o#d(1!a)]2
RS
Det"
(a!1)
'0,
Rx
a
1
(16)
46
T. Russell, A. Zecevic / Journal of Economic Dynamics & Control 24 (2000) 39}62
where RS/Rx is evaluated at the steady state using Eq. (14). Such a formulation
1
allows for an easy identi"cation of parameters for which indeterminacy can
occur. We should point out, however, that the stability of Eq. (15) only guarantees local stability of Eq. (13) around the origin. In other words, since Eq. (15) is
a local approximation, all that can be said based on the eigenvalues of the
Jacobian is that indeterminacy exists in a neighborhood of the origin.
3. The Lyapunov method
To obtain an explicit bound on initial conditions that lead to indeterminacy,
it is necessary to replace linearization with a more sophisticated approach. In
this section we will use Lyapunov's method to estimate a region of stability for
this model. The following well known result provides a basis for our analysis
(e.g. Rouche et al., 1977).
Theorem 1. Let x(t;x ) denote the solution of the nonlinear dynamic system (12)
0
that corresponds to initial condition x(0)"x , and let XLRn be a set contain0
ing the origin. Assume also that there exists a continuously di!erentiable
function < : XPR satisfying